Simplifying surds

A surd is a square root that isn’t a whole number, such as $\sqrt2$ or $\sqrt{12}$. Its decimal never ends. So questions ask you to leave answers “in surd form”, which keeps them exact.

The one rule behind everything:

Simplifying: take out the largest square

$\sqrt{72}$ is the side of a square of area 72. $72 = 36 \times 2$, so the square splits into $6 \times 6$ small squares, each of area 2. Each small square has side $\sqrt2$, so the big side is 6 lots of $\sqrt2$: $\sqrt{72} = 6\sqrt2$.

In symbols:

• 36 is the largest square factor of 72: ${\sqrt{72} = \sqrt{36 \times 2}}$;
• split the root: ${\sqrt{36} \times \sqrt2}$;
• ${\sqrt{36} = 6}$, so ${\sqrt{72} = 6\sqrt2}$.

Adding and subtracting: only like surds

$2\sqrt3 + 5\sqrt3 = 7\sqrt3$, just like $2x + 5x = 7x$. But $\sqrt2 + \sqrt3$ can’t be combined, and it is not $\sqrt5$. So simplify every surd first. Often they turn out to be multiples of the same surd.

Multiplying

Multiply the numbers outside together, and the numbers inside together. Then simplify:

• outside: ${2 \times 5 = 10}$; inside: ${3 \times 6 = 18}$. So ${2\sqrt3 \times 5\sqrt6 = 10\sqrt{18}}$;
• simplify: ${\sqrt{18} = 3\sqrt2}$, so this is ${10 \times 3\sqrt2 = 30\sqrt2}$.

Also, $\sqrt a \times \sqrt a = a$.

Brackets expand as usual. A useful pair is the difference of two squares:

The surds cancel and a whole number is left.

The square root of a surd expression

Squaring a difference of surds gives a whole number and a surd. Using $(a - b)^2 = a^2 + b^2 - 2ab$:

So to find $\sqrt{8 - 2\sqrt{15}}$, work backwards. Look for two numbers that add to 8 and multiply to 15. They are 5 and 3, so the root is $\sqrt5 - \sqrt3$.

Surds in other topics

Surds turn up whenever an answer must be exact: in trigonometry with exact values↺ such as $\sin 60^\circ = \frac{\sqrt3}{2}$, in Pythagoras, and in equations whose solution isn’t a whole number. The same rules apply.

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